benoit mandelbrot

after having mentioned his name left and right throughout this book, i will finally introduce Mandelbrot, principally as the first person with an academic title with whom i ever spoke about randomness without feeling defrauded. other mathematicians of probability would throw at me theorems with russian names such as “sobolev,” “kolmogorov,” wiener measure, without which they were lost; they had a hard time getting to the heart of the subject or exiting their little box long enough to consider its empirical flaws. with Mandelbrot, it was different: it was as if we both originated from the same country, meeting after years of frustrating exile, and were finally able to speak in our mother tongue without straining.

Fractal is a word Mandelbrot coined to describe the geometry of the rough and broken—from the Latin fractus, the origin of fractured. Fractality is the repetition of geometric patterns at different scales, revealing smaller and smaller versions of themselves. Small parts resemble, to some degree, the whole. I will try to show in this chapter how the fractal applies to the brand of uncertainty that should bear Mandelbrot’s name: Mandelbrotian randomness.

The veins in leaves look like branches; branches look like trees; rocks look like small mountains. There is no qualitative change when an object changes size.

This character of self-affinity implies that one deceptively short and simple rule of iteration can be used, either by a computer or, more randomly, by Mother Nature, to build shapes of seemingly great complexity. This can come in handy for computer graphics, but, more important, it is how nature works. Mandelbrot designed the mathematical object now known as the Mandelbrot set, the most famous object in the history of mathematics. It became popular with followers of chaos theory because it generates pictures of ever increasing complexity by using a deceptively minuscule recursive rule; recursive means that something can be reapplied to itself infinitely. You can look at the set at smaller and smaller resolutions without ever reaching the limit; you will continue to see recognizable shapes. The shapes are never the same, yet they bear an affinity to one another, a strong family resemblance.

Fractals initially made Benoît M. a pariah in the mathematical establishment.

how fitting. pariah: outcast

So Mandelbrot spent time as an intellectual refugee at an IBM research center in upstate New York. It was a f*** you money situation, as IBM let him do whatever he felt like doing.

the luxury to do whatever you want. imagine 7 billion people with that (money absent) access/privilege. nclb redefined. 100% google for 100% people. 24/7.

The computer age helped him become one of the most influential mathematicians in history, in terms of the applications of his work, way before his acceptance by the ivory tower. We will see that, in addition to its universality, his work offers an unusual attribute: it is remarkably easy to understand.

Mandelbrot came to France from Warsaw in 1936, at the age of twelve. Owing to the vicissitudes of a clandestine life during Nazi-occupied France, he was spared some of the conventional Gallic education with its uninspiring algebraic drills, becoming largely self-taught.

..in the generation of visual intuitions lies a dialectic between the mathematician and the objects generated.

The rug at eye level corresponds to Mediocristan and the law of large numbers: I am seeing the sum of undulations, and these iron out. This is like Gaussian randomness: the reason my cup of coffee does not jump is that the sum of all of its moving particles becomes smooth. Likewise, you reach certainties by adding up small Gaussian uncertainties: this is the law of large numbers.

The key here is that the fractal has numerical or statistical measures that are (somewhat) preserved across scales—the ratio is the same, unlike the Gaussian.

Fractals should be the default, the approximation, the framework. They do not solve the Black Swan problem and do not turn all Black Swans into predictable events, but they significantly mitigate the Black Swan problem by making such large events conceivable. (It makes them gray.

Recall that nonlinear processes have greater degrees of freedom than linear ones.

perfect. no?

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feb – ted2010

Fractals and the art of roughness

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Benoît B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born, French and American mathematician, noted for developing a “theory of roughness” and “self-similarity” in nature and the field of fractal geometry to help prove it, which included coining the word “fractal”. He later discovered the Mandelbrot set of intricate, never-ending fractal shapes, named in his honor.

When he was a child, his family immigrated to France in 1936. After World War II ended in 1945, Mandelbrot studied mathematics, graduating from universities in Paris and the U.S., receiving a masters degree in aeronautics from Caltech. He spent most of his career in both the U.S. and France, having dualFrench and American citizenship. In 1958 he began working for IBM, where he stayed for 35 years and was an IBM Fellow.